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Lesson Menu Five-Minute Check (over Lesson 7–2) CCSS Then/Now New Vocabulary Key Concept: Logarithm with Base b Example 1: Logarithmic to Exponential Form Example 2: Exponential to Logarithmic Form Example 3: Evaluate Logarithmic Expressions Key Concept: Parent Function of Logarithmic Functions Example 4: Graph Logarithmic Functions Key Concept: Transformations of Logarithmic Functions Example 5: Graph Logarithmic Functions Example 6: Real-World Example: Find Inverses of Exponential Functions

Over Lesson 7–2 5-Minute Check 1 Solve 4 2x = 16 3x – 1. A.x = –1 B.x = C.x = 1 D.x = 2 __ 1 2

Over Lesson 7–2 5-Minute Check 2 A.x = 10 B.x = 8 C.x = 6 D.x = 4 Solve 8 x – 1 = 2 x + 9.

Over Lesson 7–2 5-Minute Check 3 A.x < 6 B.x < 5 C.x < 4 D.x > 3 Solve 5 2x – 7 < 125.

Over Lesson 7–2 5-Minute Check 4 A.x ≥ 2 B.x ≥ 1 C.x > 0 D.x ≥ –2 Solve

Over Lesson 7–2 5-Minute Check 5 A.$18, B.$15, C.$15, D.$14, A money market account pays 5.3% interest compounded quarterly. What will be the balance in the account after 5 years if $12,000 is invested?

Over Lesson 7–2 5-Minute Check 6 A.$146,250 B.$271,250 C.$389, D.$393, Charlie borrowed $125,000 for his small business at a rate of 3.9% compounded annually for 30 years. At the end of the loan, how much will he have actually paid for the loan?

CCSS Content Standards F.IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f (kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Mathematical Practices 6 Attend to precision.

Then/Now You found the inverse of a function. Evaluate logarithmic expressions. Graph logarithmic functions.

Vocabulary logarithm logarithmic function

Concept

Example 1 Logarithmic to Exponential Form A. Write log 3 9 = 2 in exponential form. Answer: 9 = 3 2 log 3 9 = 2 → 9 = 3 2

Example 1 Logarithmic to Exponential Form Answer: B. Write in exponential form.

Example 1 A.8 3 = 2 B.2 3 = 8 C.3 2 = 8 D.2 8 = 3 A. What is log 2 8 = 3 written in exponential form?

Example 1 B. What is –2 written in exponential form? A. B. C. D.

Example 2 Exponential to Logarithmic Form A. Write 5 3 = 125 in logarithmic form. Answer: log = = 125 → log = 3

Example 2 Exponential to Logarithmic Form B. Write in logarithmic form. Answer:

Example 2 A.log 3 81 = 4 B.log 4 81 = 3 C.log 81 3 = 4 D.log 3 4 = 81 A. What is 3 4 = 81 written in logarithmic form?

Example 2 B. What is written in logarithmic form? A. B. C. D.

Example 3 Evaluate Logarithmic Expressions Evaluate log log 3 243= yLet the logarithm equal y. 243= 3 y Definition of logarithm 3 5 = 3 y 243 = 3 5 5= yProperty of Equality for Exponential Functions Answer: So, log = 5.

Example 3 Evaluate log A. B.3 C.30 D.10,000

Concept

Example 4 Graph Logarithmic Functions A. Graph the function f(x) = log 3 x. Step 1Identify the base. b = 3 Step 2Determine points on the graph. Step 3Plot the points and sketch the graph. Because 3 > 1, use the points (1, 0), and (b, 1).

Example 4 Graph Logarithmic Functions (1, 0) (b, 1) → (3, 1) Answer:

Example 4 Graph Logarithmic Functions Step 1Identify the base. B. Graph the function Step 2Determine points on the graph.

Example 4 Graph Logarithmic Functions Step 3Sketch the graph. Answer:

Example 4 A. Graph the function f(x) = log 5 x. A. B. C.D.

Example 4 B. Graph the function. A. B. C.D.

Concept

Example 5 Graph Logarithmic Functions This represents a transformation of the graph f(x) = log 6 x. ● : The graph is compressed vertically. ● h = 0: There is no horizontal shift. ● k = –1: The graph is translated 1 unit down.

Example 5 Graph Logarithmic Functions Answer:

Example 5 Graph Logarithmic Functions ● |a| = 4: The graph is stretched vertically. ● h = –2: The graph is translated 2 units to the left. ● k = 0: There is no vertical shift.

Example 5 Graph Logarithmic Functions Answer:

Example 5 A. B. C.D.

Example 5 A. B. C.D.

Example 6 Find Inverses of Exponential Functions A. AIR PRESSURE At Earth’s surface, the air pressure is defined as 1 atmosphere. Pressure decreases by about 20% for each mile of altitude. Atmospheric pressure can be modeled by P = 0.8 x, where x measures altitude in miles. Find the atmospheric pressure in atmospheres at an altitude of 8 miles. P= 0.8 x Original equation = Substitute 8 for x. ≈ 0.168Use a calculator. Answer: atmosphere

Example 6 Find Inverses of Exponential Functions B. AIR PRESSURE At Earth’s surface, the air pressure is defined as 1 atmosphere. Pressure decreases by about 20% for each mile of altitude. Atmospheric pressure can be modeled by P = 0.8 x, where x measures altitude in miles. Write an equation for the inverse of the function. P= 0.8 x Original equation x= 0.8 P Replace x with P, replace P with x, and solve for P. P= log 0.8 xDefinition of logarithm Answer: P = log 0.8 x

Example 6 A.42 B.41 C.40 D.39 A. AIR PRESSURE The air pressure of a car tire is 44 lbs/in 2. The pressure decreases gradually by about 1% for each trip of 50 miles driven. The air pressure can be modeled by P = 44(0.99 x ), where x measures the number of 50-mile trips. Find the air pressure in pounds per square inch after driving 350 miles.

Example 6 B. AIR PRESSURE The air pressure of a car tire is 44 lbs/in 2. The pressure decreases gradually by about 1% for each trip of 50 miles driven. The air pressure can be modeled by P = 44(0.99 x ), where x measures the number of 50-mile trips. Write an equation for the inverse of the function. A.P = 44 log 0.99 x B.P = log 0.99 x C. D.

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